A different rule applies on that one. The batter (the trailing runner)
would be out for passing the lead runner.
There's no need for the trailing runner to actually touch the next base before the lead runner does. The instant she passes the lead runner she is out.
Now it gets a little bit trickier...
If there were less than two outs, the offense scores one run on this play (the lead runner, assuming that she properly completes her base running duties to home).
If there are already two outs...
If the pass happened BEFORE first base, then no runs score. This would be regarded as the third out being made by the batter-runner before safely reaching first base.
If the pass happened AFTER first base, then one run would score (the lead runner- again, assuming she completes her baserunning duties to home).